I just tried something like this , Check it once.
Create a parameter , Name it Start Date , Select Data type as Date , Select Allowable values to Range , then Use Set from filed option to give min and max dates from your required filed to filter. Select Step size as Months. For current value option select starting of the any month and Display format as Automatic. In Edit Parameters option select Type in option.
Create a calculation to relate this parameter with your required filter filed and use this calculation in filter shelf.
This will be useful for start date filter.
Note : from this parameter when we select from day 1 to 16 of any month it will take the starting day of that month , but when we select from day 17 to last day of any month it will take the starting day of next month.
My requirement From date always first day of month and End date should be always last day of month or next month 1 st day.